Properties

Label 4-1148e2-1.1-c1e2-0-11
Degree $4$
Conductor $1317904$
Sign $1$
Analytic cond. $84.0307$
Root an. cond. $3.02767$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 3·7-s − 2·9-s − 3·11-s + 4·16-s − 8·25-s + 6·28-s − 12·29-s + 4·36-s + 2·37-s + 3·43-s + 6·44-s + 2·49-s − 7·53-s + 6·63-s − 8·64-s + 3·67-s − 24·71-s + 9·77-s + 27·79-s − 5·81-s + 6·99-s + 16·100-s − 24·107-s − 24·109-s − 12·112-s − 36·113-s + ⋯
L(s)  = 1  − 4-s − 1.13·7-s − 2/3·9-s − 0.904·11-s + 16-s − 8/5·25-s + 1.13·28-s − 2.22·29-s + 2/3·36-s + 0.328·37-s + 0.457·43-s + 0.904·44-s + 2/7·49-s − 0.961·53-s + 0.755·63-s − 64-s + 0.366·67-s − 2.84·71-s + 1.02·77-s + 3.03·79-s − 5/9·81-s + 0.603·99-s + 8/5·100-s − 2.32·107-s − 2.29·109-s − 1.13·112-s − 3.38·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1317904\)    =    \(2^{4} \cdot 7^{2} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(84.0307\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1317904} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1317904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + p T^{2} \)
7$C_2$ \( 1 + 3 T + p T^{2} \)
41$C_2$ \( 1 + T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \)
47$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 125 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 - 12 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2^2$ \( 1 + 65 T^{2} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.81984255136903205261579808521, −7.14867232559295101187766231845, −6.64447309479564266365116059662, −6.05677072582506833294348520457, −5.71837803290602095684702552837, −5.44463946581307225410414319337, −4.95635577557968888449209561031, −4.20987702177701571378651861534, −3.87790605820721841521259398882, −3.40322535550169985950590696955, −2.85785765538364129691072107754, −2.28984803304305741296664810867, −1.41170618977347961541338413914, 0, 0, 1.41170618977347961541338413914, 2.28984803304305741296664810867, 2.85785765538364129691072107754, 3.40322535550169985950590696955, 3.87790605820721841521259398882, 4.20987702177701571378651861534, 4.95635577557968888449209561031, 5.44463946581307225410414319337, 5.71837803290602095684702552837, 6.05677072582506833294348520457, 6.64447309479564266365116059662, 7.14867232559295101187766231845, 7.81984255136903205261579808521

Graph of the $Z$-function along the critical line